Standard Scores

by

Adam J. McKee, Ph.D.

January, 2004

                In order to use the characteristics of the normal curve to help us determine the probability of an occurrence of a given value, we must convert the raw scores to standardized or z-scores.  That is, we must convert the score into standard deviation units.  z can be calculated using the following formula:

Where:

X = raw score
X bar = mean
s = standard deviation

If you have a GPA of 3.5 and we know the mean of GPAs to be 2.93 and the standard deviation to be .33, we can calculate z as follows:

From this we can say that a GPA of 3.5 lies 1.73 standard deviations above the mean.

                A special distribution goes along with z-scores.  It is known as the standard normal distribution and has a mean of zero and a SD of one.  Note that the normal curve is defined by the dispersion (variance) and centrality (mean) of the data, but the values for these measures are infinitely many. The centrality determines whether the distribution is symmetrical or not, and dispersion of data points from the mean causes the bell shape.  This distribution lets us determine where z-scores fall under the curve by looking at table that lists proportions under the curve for every possible z-score.  Thus, we can answer the following questions with z-scores:

1.  What proportion of scores fall between the mean and a given raw score?

2.  What proportion of scores fall above or below a given raw score?

3.  What proportion of scores fall between two raw scores?

4.  What raw score falls above or below a given percentage score?

                Remember that the curve is symmetrical, so signs can be ignored when determining distance from the mean.  It is usually best to draw the curve and shade in the areas that interest you.  This is especially true when you are attempting to define the proportion of scores between two raw scores, because the table values will usually not directly answer the question.   Other standard scores can be easily calculated, but there is seldom a need for this in Criminal Justice research.  The practice is more common in educational testing.

                Note that percentile ranks are a function of the normal curve as well.  If you graduated in the top ten percent of your class, then you are said to be in the 90th percentile.  This is the point where 90% of scores fall below.  Percentiles are often used to establish an idea of who the cream of the crop is in educational settings.   National merit scholarships are awarded to those who score in the 99th percentile on the PSAT.  Soldiers graduating basic training at a certain percentile are often invited to attend officer candidate school.  Note that there cannot be a 100th percentile.  That would mean that all scores fell below that one, which is impossible.

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